Donovan's conjecture over fields and complete mixed-characteristic DVRs

Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.

Manuscripts (2)

Donovan's Conjecture over Algebraically Closed Fields

September 24, 2026 · source · cite as \cite{OAI:Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026}

We prove Donovan's conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.

Integral Donovan Finiteness over Witt Vectors

September 25, 2026 · source · cite as \cite{OAI:Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026}

We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over W(\overline{\mathbb F}_p). The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.

Result 203 · Algebra. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.